Splines of complex order, fractional operators and applications in signal and image processing

Schoenberg's splines have found their way into a large number of applied mathematical areas, for example,signal processing, computer graphics, computer assisted geometric design, the numerics of partial differentialequations and many more. With their flexible parameters they are adaptable to the particular task at hand andbecause of their simple representation they are easily implementable. These are properties that are imperativefor a successful application. In addition, as splines are piecewise polynomials, they facilitate the interpretationof numerical results.

In recent years, splines of fractional and complex order were defined, which, due to the existence of additionalparameters, are even more adaptable. The continuous order parameter allows a precise adjustment to theregularity of the problem and the complex degree the extraction of phase and amplitude. These splinesgenerate new mathematical interconnections ranging from fractional differential operators, Dirichlet averages,Riesz transformations and phase in higher dimensions, to curvature detection operators for image analysis.

For concrete applications, the complete theoretical and numerical analyses of the approximation-theoreticproperties of the spaces generated by splines of complex order are still missing. We want to close this gap byproviding a comprehensive analysis of splines of complex order. For this purpose, we will emphasize both thetheoretical results and numerical analysis, as well as the algorithmic implementation.

Principal Investigator(s) at the University	Prof. Dr. Brigitte Forster-Heinlein (Professur für Angewandte Mathematik)
Project period	01.01.2024 - 31.12.2016
Verlängert bis:	28.02.2018
Website	https://www.fim.uni-passau.de/angewandte-mathematik/forschung
Source of funding	DFG - Deutsche Forschungsgemeinschaft > DFG - Sachbeihilfe